Basic Concepts of Graph Algorithms

Transcription

1 Basic Concepts of Graph Algorithms Combinatorics for Computer Science (Units 6 and 7) S. Gill Williamson

2 c S. Gill Williamson 2012

3 Preface From 1970 to 1990 I ran a graduate seminar on algebraic and algorithmic combinatorics in the Department of Mathematics, UCSD. From 1972 to 1990 algorithmic combinatorics became the principal topic. The seminar notes from 1970 to 1985 were combined and published as a book, Combinatorics for Computer Science (CCS), published by Computer Science Press. Each of the units of study from the seminar became a chapter in this book. Here, we isolate a combined Unit 6 and Unit 7, corresponding to Chapter 6 and Chapter 7 of CCS, and reconstruct the original very helpful unit specific index associated with these two units. Theorems, figures, etc., are numbered sequentially: DEFINITION 6.10 and EXERCISE 6.29 refer to numbered items 10 and 29 of Unit 6 (or Chapter 6 in CCS). Unit 6 contains basic material at an introductory level. Unit 7 applies Unit 6 to a more advanced topic (planarity testing). These notes focus on the visualization of algorithms through the use of graphical and pictorial methods. This approach is both fun and powerful, preparing you to invent your own algorithms for a wide range of problems. For further references and ongoing research, search the Web, particularly Wikipedia and the mathematics arxiv (arxiv.org). Also available in this series are Basic Concepts of Linear Order (Unit 1), Sorting and Listing (Unit 2 and Unit 3)), and Pólya Counting Theory (Unit 4). Units 6 and 7 are essentially independent of earlier units. The exercises in this material were designed for student presentation in the seminar. In many cases, these presentations were done after we had gone through the entire unit. A good strategy is to read and understand these exercises and return to the ones that interest you after you have read the unit. S. Gill Williamson, iii

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5 Table of Contents Unit 6: Basic Concepts of Graph algorithms...1 Unit 7: Depth First Search and Planarity Subject Index v

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7 Unit 6 Basic Concepts of Graph Algorithms 1

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72 Unit 7 Depth First Search and Planarity 66

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103 Index procedure BREAK(H), CONSTRUCT T AND B, 70 ESEQ(T), 6 INVPRU(T), 8 JOIN(H), MAXSON, 35 MOVE(t, δ, α, β), 18 NEXT(T i ), 23 NEXTPERM(A, m), 20 PATH(e), 78 PRU(T), 7 SLOPPY PLANARITY TEST, construct lineal span. tree, 28 acyclic no cycles, 5 adjacency table, 4, 30, 77 adjacent to a VERT(b), 30 articulation points graph, 50, 51 automorphism group graph, 40 backedges vs. chords, 68 bicomponent tree, bicomponents complexity issues, 73 computing, 72, 73 biconnected articulation points, 51 biconnected graph, 47 BIJ(A,B) 97 bijections A to B, bijections BIJ(A,B), binary Gray code, 21 breadth first, edges BRE(T), 13 breadth first, vertices BRV(T), 13 bridge SEG(e), 80 bridge carrier, 80 bridge graph, bridges cycle, 52, 53 subgraph, 53 canonical diagrams trees, 43 carrier of bridge, 80 in cycle, 53, 54 components connected, 5 connected graph, 5 has lineal spanning tree, 25 lineal spanning tree, 24 connected graphs on V CON(V), 36 connected graphs, four vertices, 37 consistent embedding, 80 contents, v cycle bridges of, 52, 53 carrier of bridge, 53, 54 length n, 5 cycle bicomponent, 53 cycle equivalent edges, 50

104 cycle tree, 56, 57 data structure terminal, 33 depth first edges, DFE(T), 11, 66, 67 vertices, DFV(T), 11, 66, 67 depth first sequences definition, 11 direct access model data structure, 28, 29 directed graph, 2 doubly linked list pointers, 30 edge directed, 6 incident on vertex, 6 loop, 4 natural directed, 9 sequence, ESEQ(T), 6 stack in tree, STACK(e), 12 edge action EPER(V) on PER(P 2 (V)), edge set, 2 edges backedges, 68 breadth first BRE(T), 13 chords, 68 cycle equivalent, 50 postorder POSE(T), 12, preorder PREE(T), 12, 69 71, 74 embedding, 2 consistent, 80 preorder vertices, 75, 76 properly ordered, 76 EPER(V) action on GRAPHS(V), graph 2-connected or biconnected, 47 articulation points, 50, 51 automorphism stabilizer, 40 bridge graph, chain of edges, 10 connected, 5 defined as triple, 2 directed, 2, 4 directed path, 10 embedding of, intuitive idea, 1 ordered, 4, 66 planar, planar embedding, rank, nullity, 47, 48 rooted lineal spanning tree, 25 rooted lineal subtree, 25 tree definition, 5 undirected UND(G), 10 graphs isomorphism of, greedy algorithm, 50 Hopcroft, 96 hypergraph, 3 incidence function, 2 inversion enumerator polynomial, 36 inversions definition, 35 isomorphic graphs, isomorphism of graphs, 38 Kuratowski subgraphs, 63 lineal spanning tree, 24, 68, 69 algorithm for, 27 chord, backedge, 34 complexity, connected graph, 25 98

105 first proof of, 26 inversions in, 35 second proof of, 27 lineal subtree, 25 connected graph, 24, 68 ordered, 24, 68 loop, 2, 4 LOW1, LOW2 values, 72, 73 multigraph, 3 multiset, 3 notation n = {1,..., n}, 6 ordered rooted tree, 42, 66 orderly algorithm, 38, 43 basic mapping B, 44 computing B 1, 44 example, exercises, 46, 47 partition set unordered Π(n, k), 17 path length n, 4 path directed, 10 path tree, 78 example, 79 PER(A) permutations of A, PER(V) permutations of vertices, planar bipartite BRGR(C), planar bipartite if e 2v 4, 62, 63 planar embedding Euler e v+r = 2, 63 planar embedding graph, planar graph, planar if e 3v 6, 62, 63 planar not consistent, 81 planarity test outline, planarity testing MAX MIN CHAIN, 94 directly linked test, 91 example, non-bichromatic SEGGR, 95 relevant SEGGR(e, H), 92, 93 pointers, polynomial inversion enumerator, 36 postorder edges, POSE(T), 12 postorder vertices, POSV(T), 12 Prüffer sequence of T, 7 preface, iii preorder edges, PREE(T), 12 preorder vertices, PREV(T), 12 principal subtree, 66 definition, 11 rank of G edges in spanning forest, 47 segment graph bichromatic, 82 planarity condition, 82 spanning forest, 24, 68 subgraph, 24, 68 tree, 24, 68 tree - lineal, 24, 68 tree - rooted, 24, 68 spanning forest chords, 24, 68 rank of G, 47 spanning subgraph, 47 stability subgroup graph, 40 stack STACK(e ), 24 stack in tree, edge, STACK(e), 13 99

106 stack in tree, vertex, STACK(x), 13 subgraph bridges of, 53 spanning, 47 subset action of PER(A), subsets of A: P(A) size 2: P 2 (A), subtree lineal, 25 Tarjan, 96 Towers of Hanoi, tree breadth first order, 12 definition, 5 distance between vertices, 9 lineal spanning, 24, 68 natural directed, 9, 66 of cycles, 56, 57 of paths, ordered rooted, 42, 66 ordered rooted ORTR, 9, 66 rooted RTR, 9, 66 subtree, 24, 68 unique path to root, 9 trees all TR(V), 7 ancestor of y, 22, 68 canonical diagrams, 43 descendant of x, 22, 68 edge initial vertex IN(e), 22 terminal vertex TM(e), 22 father of vertex, 22 lexicographic order, 9 lineal descendants, 22, 68 notation, 21 son of vertex, 22 trees rooted at v TR(V, v), 36 vector notation L, L, x L, etc., 22 vertex adjacent, 4, 5, 66 degree of, 6 pendant, leaf, terminal, 6 sequence, 4 stack in tree, STACK(x), 12 vertex set, 2 vertices breadth first BRV(T), 13 internal of tree INT(T), 8 path equivalence, 5 pendant of tree, PEND(T), 6 postorder POSV(T), 12, preorder PREV(T), 12, 69 71,